Mixing
The equality in Descartes' rule of signs
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The standard formulation of Descartes' rule of signs gives upper bounds for the number of positive roots of a real polynomial. However, in the Wikipedia article devoted to it https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs a "Special case" is mentioned without any proof or reference:
"The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case. "
Does anyone know any reference for this statement?
abstract-algebra algebra-precalculus
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
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add a comment |
$begingroup$
The standard formulation of Descartes' rule of signs gives upper bounds for the number of positive roots of a real polynomial. However, in the Wikipedia article devoted to it https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs a "Special case" is mentioned without any proof or reference:
"The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case. "
Does anyone know any reference for this statement?
abstract-algebra algebra-precalculus
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
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I never noticed this (apparently useful) special case (+1)
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– Peter
Jan 26 at 13:47
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How about the article linked in this answer.
$endgroup$
– jgon
Jan 26 at 18:59
add a comment |
$begingroup$
The standard formulation of Descartes' rule of signs gives upper bounds for the number of positive roots of a real polynomial. However, in the Wikipedia article devoted to it https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs a "Special case" is mentioned without any proof or reference:
"The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case. "
Does anyone know any reference for this statement?
abstract-algebra algebra-precalculus
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
$endgroup$
The standard formulation of Descartes' rule of signs gives upper bounds for the number of positive roots of a real polynomial. However, in the Wikipedia article devoted to it https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs a "Special case" is mentioned without any proof or reference:
"The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and negative roots. Since it is easy to determine the multiplicity of zero as a root, the sign of all roots can be determined in this case. "
Does anyone know any reference for this statement?
abstract-algebra algebra-precalculus
abstract-algebra algebra-precalculus
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
asked Jan 26 at 13:20
Antonio Jesús Ureña AlcázarAntonio Jesús Ureña Alcázar
61
61
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I never noticed this (apparently useful) special case (+1)
$endgroup$
– Peter
Jan 26 at 13:47
$begingroup$
How about the article linked in this answer.
$endgroup$
– jgon
Jan 26 at 18:59
add a comment |
$begingroup$
I never noticed this (apparently useful) special case (+1)
$endgroup$
– Peter
Jan 26 at 13:47
$begingroup$
How about the article linked in this answer.
$endgroup$
– jgon
Jan 26 at 18:59
$begingroup$
I never noticed this (apparently useful) special case (+1)
$endgroup$
– Peter
Jan 26 at 13:47
$begingroup$
I never noticed this (apparently useful) special case (+1)
$endgroup$
– Peter
Jan 26 at 13:47
$begingroup$
How about the article linked in this answer.
$endgroup$
– jgon
Jan 26 at 18:59
$begingroup$
How about the article linked in this answer.
$endgroup$
– jgon
Jan 26 at 18:59
add a comment |
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$begingroup$
I never noticed this (apparently useful) special case (+1)
$endgroup$
– Peter
Jan 26 at 13:47
$begingroup$
How about the article linked in this answer.
$endgroup$
– jgon
Jan 26 at 18:59